Geometric Progression Quiz

$a = 3$, $r = 2$. $a_5 = 3 \\times 2^{4} = 3 \\times 16 = 48$.

$S_4 = \\frac{a(r^n - 1)}{r - 1} = \\frac{2(3^4 - 1)}{3 - 1} = \\frac{2(80)}{2} = 80$.

$a = 1$, $r = 1/3$. $S_\\infty = \\frac{a}{1-r} = \\frac{1}{1-1/3} = \\frac{1}{2/3} = \\frac{3}{2}$.

$a_6/a_3 = r^3 = 96/12 = 8 \\Rightarrow r = 2$. $a_3 = a \\times 4 = 12 \\Rightarrow a = 3$.

$ar^3=24$, $ar^6=192$. $r^3=192/24=8 \\Rightarrow r=2$. $a(8)=24 \\Rightarrow a=3$.

$a=3$, $r=2$. $S_6 = 3(2^6-1)/(2-1) = 3\\times63 = 189$.

$S_\\infty = a/(1-r) \\Rightarrow 20 = 5/(1-r) \\Rightarrow 1-r = 5/20 = 0.25 \\Rightarrow r = 0.75$.

Let $4, ar, ar^2, 32$ be GP. $4r^3=32 \\Rightarrow r^3=8 \\Rightarrow r=2$. Means: $4\\times2=8$, $8\\times2=16$.

$ar^{n-1} = 128 \\Rightarrow 2(2)^{n-1} = 128 \\Rightarrow 2^{n-1} = 64 = 2^6 \\Rightarrow n-1=6 \\Rightarrow n=7$.

Product $= a^5 r^{0+1+2+3+4} = 2^5 r^{10} = 2^5\\times3^{10}$.